I always wonder if keeping the car turned on with the heater blowing and oil and water circulating is better than just shutting everything down

With our modern cars, I would suggest to turn them off. The fan may be turning, but is only cooling the coolant (which is not being circulated because of the broken water pump) If you heater were still working, then it wouldn't be bad to let it run idle for maximum 1 minute. The plastic cover over the engine avoids any air cooling anyways. With a running oil pump, the oil would still "transport" the heat away from the hottest part (the cylinder head), but as soon as you shut off, there should not be more heat produced.

When this happened to me, I contemplated whether or not to re-start my engine and run it for 20 seconds after 3-5 min of cooling. But I didn't. I wasn't sure if that would really make a difference. The heat created by the combustion of gasses would heat up your cylinder heads again, and the amount of cooler oil pump around the cylinders would be very little...